Sr Examen
Integral de x^2/(x^4+1)^2 dx
Solución
Ha introducido
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1 / | | 2 | x | --------- dx | 2 | / 4 \ | \x + 1/ | / 0
$$\int\limits_{0}^{1} \frac{x^{2}}{\left(x^{4} + 1\right)^{2}}\, dx$$
Integral(x^2/(x^4 + 1)^2, (x, 0, 1))
Respuesta (Indefinida)
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/ | | 2 3 ___ / 2 ___\ ___ / ___\ ___ / ___\ ___ / 2 ___\ | x x \/ 2 *log\1 + x + x*\/ 2 / \/ 2 *atan\1 + x*\/ 2 / \/ 2 *atan\-1 + x*\/ 2 / \/ 2 *log\1 + x - x*\/ 2 / | --------- dx = C + -------- - --------------------------- + ----------------------- + ------------------------ + --------------------------- | 2 4 32 16 16 32 | / 4 \ 4 + 4*x | \x + 1/ | /
$$\int \frac{x^{2}}{\left(x^{4} + 1\right)^{2}}\, dx = C + \frac{x^{3}}{4 x^{4} + 4} + \frac{\sqrt{2} \log{\left(x^{2} - \sqrt{2} x + 1 \right)}}{32} - \frac{\sqrt{2} \log{\left(x^{2} + \sqrt{2} x + 1 \right)}}{32} + \frac{\sqrt{2} \operatorname{atan}{\left(\sqrt{2} x - 1 \right)}}{16} + \frac{\sqrt{2} \operatorname{atan}{\left(\sqrt{2} x + 1 \right)}}{16}$$
Gráfica
Respuesta
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___ / ___\ ___ ___ / ___\ 1 \/ 2 *log\2 + \/ 2 / pi*\/ 2 \/ 2 *log\2 - \/ 2 / - - -------------------- + -------- + -------------------- 8 32 32 32
$$- \frac{\sqrt{2} \log{\left(\sqrt{2} + 2 \right)}}{32} + \frac{\sqrt{2} \log{\left(2 - \sqrt{2} \right)}}{32} + \frac{1}{8} + \frac{\sqrt{2} \pi}{32}$$
=
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___ / ___\ ___ ___ / ___\ 1 \/ 2 *log\2 + \/ 2 / pi*\/ 2 \/ 2 *log\2 - \/ 2 / - - -------------------- + -------- + -------------------- 8 32 32 32
$$- \frac{\sqrt{2} \log{\left(\sqrt{2} + 2 \right)}}{32} + \frac{\sqrt{2} \log{\left(2 - \sqrt{2} \right)}}{32} + \frac{1}{8} + \frac{\sqrt{2} \pi}{32}$$
1/8 - sqrt(2)*log(2 + sqrt(2))/32 + pi*sqrt(2)/32 + sqrt(2)*log(2 - sqrt(2))/32
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.